https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Cosmicgenius&feedformat=atom AoPS Wiki - User contributions [en] 2021-05-10T09:32:57Z User contributions MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=2016_AIME_II_Problems&diff=148432 2016 AIME II Problems 2021-03-04T01:16:29Z <p>Cosmicgenius: /* Problem 13 */</p> <hr /> <div>{{AIME Problems|year=2016|n=II}}<br /> ==Problem 1==<br /> Initially Alex, Betty, and Charlie had a total of &lt;math&gt;444&lt;/math&gt; peanuts. Charlie had the most peanuts, and Alex had the least. The three numbers of peanuts that each person had formed a geometric progression. Alex eats &lt;math&gt;5&lt;/math&gt; of his peanuts, Betty eats &lt;math&gt;9&lt;/math&gt; of her peanuts, and Charlie eats &lt;math&gt;25&lt;/math&gt; of his peanuts. Now the three numbers of peanuts each person has forms an arithmetic progression. Find the number of peanuts Alex had initially.<br /> <br /> [[2016 AIME II Problems/Problem 1 | Solution]]<br /> <br /> ==Problem 2==<br /> There is a &lt;math&gt;40\%&lt;/math&gt; chance of rain on Saturday and a &lt;math&gt;30\%&lt;/math&gt; chance of rain on Sunday. However, it is twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday. The probability that it rains at least one day this weekend is &lt;math&gt;\frac{a}{b}&lt;/math&gt;, where &lt;math&gt;a&lt;/math&gt; and &lt;math&gt;b&lt;/math&gt; are relatively prime positive integers. Find &lt;math&gt;a+b&lt;/math&gt;.<br /> <br /> [[2016 AIME II Problems/Problem 2 | Solution]]<br /> <br /> ==Problem 3==<br /> Let &lt;math&gt;x,y,&lt;/math&gt; and &lt;math&gt;z&lt;/math&gt; be real numbers satisfying the system<br /> &lt;cmath&gt;<br /> \begin{align*}<br /> \log_2(xyz-3+\log_5 x)&amp;=5,\\<br /> \log_3(xyz-3+\log_5 y)&amp;=4,\\<br /> \log_4(xyz-3+\log_5 z)&amp;=4.<br /> \end{align*}<br /> &lt;/cmath&gt;<br /> Find the value of &lt;math&gt;|\log_5 x|+|\log_5 y|+|\log_5 z|&lt;/math&gt;.<br /> <br /> [[2016 AIME II Problems/Problem 3 | Solution]]<br /> <br /> ==Problem 4==<br /> An &lt;math&gt;a \times b \times c&lt;/math&gt; rectangular box is built from &lt;math&gt;a \cdot b \cdot c&lt;/math&gt; unit cubes. Each unit cube is colored red, green, or yellow. Each of the &lt;math&gt;a&lt;/math&gt; layers of size &lt;math&gt;1 \times b \times c&lt;/math&gt; parallel to the &lt;math&gt;(b \times c)&lt;/math&gt; faces of the box contains exactly &lt;math&gt;9&lt;/math&gt; red cubes, exactly &lt;math&gt;12&lt;/math&gt; green cubes, and some yellow cubes. Each of the &lt;math&gt;b&lt;/math&gt; layers of size &lt;math&gt;a \times 1 \times c&lt;/math&gt; parallel to the &lt;math&gt;(a \times c)&lt;/math&gt; faces of the box contains exactly &lt;math&gt;20&lt;/math&gt; green cubes, exactly &lt;math&gt;25&lt;/math&gt; yellow cubes, and some red cubes. Find the smallest possible volume of the box.<br /> <br /> [[2016 AIME II Problems/Problem 4 | Solution]]<br /> ==Problem 5==<br /> Triangle &lt;math&gt;ABC_0&lt;/math&gt; has a right angle at &lt;math&gt;C_0&lt;/math&gt;. Its side lengths are pairwise relatively prime positive integers, and its perimeter is &lt;math&gt;p&lt;/math&gt;. Let &lt;math&gt;C_1&lt;/math&gt; be the foot of the altitude to &lt;math&gt;\overline{AB}&lt;/math&gt;, and for &lt;math&gt;n \geq 2&lt;/math&gt;, let &lt;math&gt;C_n&lt;/math&gt; be the foot of the altitude to &lt;math&gt;\overline{C_{n-2}B}&lt;/math&gt; in &lt;math&gt;\triangle C_{n-2}C_{n-1}B&lt;/math&gt;. The sum &lt;math&gt;\sum_{n=2}^\infty C_{n-2}C_{n-1} = 6p&lt;/math&gt;. Find &lt;math&gt;p&lt;/math&gt;.<br /> <br /> [[2016 AIME II Problems/Problem 5 | Solution]]<br /> <br /> ==Problem 6==<br /> For polynomial &lt;math&gt;P(x)=1-\dfrac{1}{3}x+\dfrac{1}{6}x^{2}&lt;/math&gt;, define<br /> &lt;math&gt;Q(x)=P(x)P(x^{3})P(x^{5})P(x^{7})P(x^{9})=\sum_{i=0}^{50} a_ix^{i}&lt;/math&gt;.<br /> Then &lt;math&gt;\sum_{i=0}^{50} |a_i|=\dfrac{m}{n}&lt;/math&gt;, where &lt;math&gt;m&lt;/math&gt; and &lt;math&gt;n&lt;/math&gt; are relatively prime positive integers. Find &lt;math&gt;m+n&lt;/math&gt;.<br /> <br /> [[2016 AIME II Problems/Problem 6 | Solution]]<br /> ==Problem 7==<br /> Squares &lt;math&gt;ABCD&lt;/math&gt; and &lt;math&gt;EFGH&lt;/math&gt; have a common center and &lt;math&gt;\overline{AB} || \overline{EF}&lt;/math&gt;. The area of &lt;math&gt;ABCD&lt;/math&gt; is 2016, and the area of &lt;math&gt;EFGH&lt;/math&gt; is a smaller positive integer. Square &lt;math&gt;IJKL&lt;/math&gt; is constructed so that each of its vertices lies on a side of &lt;math&gt;ABCD&lt;/math&gt; and each vertex of &lt;math&gt;EFGH&lt;/math&gt; lies on a side of &lt;math&gt;IJKL&lt;/math&gt;. Find the difference between the largest and smallest positive integer values for the area of &lt;math&gt;IJKL&lt;/math&gt;.<br /> <br /> [[2016 AIME II Problems/Problem 7 | Solution]]<br /> <br /> ==Problem 8==<br /> Find the number of sets &lt;math&gt;\{a,b,c\}&lt;/math&gt; of three distinct positive integers with the property that the product of &lt;math&gt;a,b,&lt;/math&gt; and &lt;math&gt;c&lt;/math&gt; is equal to the product of &lt;math&gt;11,21,31,41,51,&lt;/math&gt; and &lt;math&gt;61&lt;/math&gt;.<br /> <br /> [[2016 AIME II Problems/Problem 8 | Solution]]<br /> <br /> ==Problem 9==<br /> The sequences of positive integers &lt;math&gt;1,a_2, a_3,...&lt;/math&gt; and &lt;math&gt;1,b_2, b_3,...&lt;/math&gt; are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let &lt;math&gt;c_n=a_n+b_n&lt;/math&gt;. There is an integer &lt;math&gt;k&lt;/math&gt; such that &lt;math&gt;c_{k-1}=100&lt;/math&gt; and &lt;math&gt;c_{k+1}=1000&lt;/math&gt;. Find &lt;math&gt;c_k&lt;/math&gt;.<br /> <br /> [[2016 AIME II Problems/Problem 9 | Solution]]<br /> ==Problem 10==<br /> Triangle &lt;math&gt;ABC&lt;/math&gt; is inscribed in circle &lt;math&gt;\omega&lt;/math&gt;. Points &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;Q&lt;/math&gt; are on side &lt;math&gt;\overline{AB}&lt;/math&gt; with &lt;math&gt;AP&lt;AQ&lt;/math&gt;. Rays &lt;math&gt;CP&lt;/math&gt; and &lt;math&gt;CQ&lt;/math&gt; meet &lt;math&gt;\omega&lt;/math&gt; again at &lt;math&gt;S&lt;/math&gt; and &lt;math&gt;T&lt;/math&gt; (other than &lt;math&gt;C&lt;/math&gt;), respectively. If &lt;math&gt;AP=4,PQ=3,QB=6,BT=5,&lt;/math&gt; and &lt;math&gt;AS=7&lt;/math&gt;, then &lt;math&gt;ST=\frac{m}{n}&lt;/math&gt;, where &lt;math&gt;m&lt;/math&gt; and &lt;math&gt;n&lt;/math&gt; are relatively prime positive integers. Find &lt;math&gt;m+n&lt;/math&gt;.<br /> <br /> [[2016 AIME II Problems/Problem 10 | Solution]]<br /> ==Problem 11==<br /> For positive integers &lt;math&gt;N&lt;/math&gt; and &lt;math&gt;k&lt;/math&gt;, define &lt;math&gt;N&lt;/math&gt; to be &lt;math&gt;k&lt;/math&gt;-nice if there exists a positive integer &lt;math&gt;a&lt;/math&gt; such that &lt;math&gt;a^{k}&lt;/math&gt; has exactly &lt;math&gt;N&lt;/math&gt; positive divisors. Find the number of positive integers less than &lt;math&gt;1000&lt;/math&gt; that are neither &lt;math&gt;7&lt;/math&gt;-nice nor &lt;math&gt;8&lt;/math&gt;-nice.<br /> <br /> [[2016 AIME II Problems/Problem 11 | Solution]]<br /> ==Problem 12==<br /> The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.<br /> <br /> &lt;asy&gt;<br /> draw(Circle((0,0), 4));<br /> draw(Circle((0,0), 3));<br /> draw((0,4)--(0,3));<br /> draw((0,-4)--(0,-3));<br /> draw((-2.598, 1.5)--(-3.4641, 2));<br /> draw((-2.598, -1.5)--(-3.4641, -2));<br /> draw((2.598, -1.5)--(3.4641, -2));<br /> draw((2.598, 1.5)--(3.4641, 2));<br /> &lt;/asy&gt;<br /> <br /> [[2016 AIME II Problems/Problem 12 | Solution]]<br /> <br /> ==Problem 13==<br /> Beatrix is going to place six rooks on a &lt;math&gt;6 \times 6&lt;/math&gt; chessboard where both the rows and columns are labeled &lt;math&gt;1&lt;/math&gt; to &lt;math&gt;6&lt;/math&gt;; the rooks are placed so that no two rooks are in the same row or the same column. The ''value'' of a square is the sum of its row number and column number. The ''score'' of an arrangement of rooks is the least value of any occupied square. The average score over all valid configurations is &lt;math&gt;\frac{p}{q}&lt;/math&gt;, where &lt;math&gt;p&lt;/math&gt; and &lt;math&gt;q&lt;/math&gt; are relatively prime positive integers. Find &lt;math&gt;p+q&lt;/math&gt;.<br /> <br /> [[2016 AIME II Problems/Problem 13 | Solution]]<br /> <br /> ==Problem 14==<br /> Equilateral &lt;math&gt;\triangle ABC&lt;/math&gt; has side length &lt;math&gt;600&lt;/math&gt;. Points &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;Q&lt;/math&gt; lie outside the plane of &lt;math&gt;\triangle ABC&lt;/math&gt; and are on opposite sides of the plane. Furthermore, &lt;math&gt;PA=PB=PC&lt;/math&gt;, and &lt;math&gt;QA=QB=QC&lt;/math&gt;, and the planes of &lt;math&gt;\triangle PAB&lt;/math&gt; and &lt;math&gt;\triangle QAB&lt;/math&gt; form a &lt;math&gt;120^{\circ}&lt;/math&gt; dihedral angle (the angle between the two planes). There is a point &lt;math&gt;O&lt;/math&gt; whose distance from each of &lt;math&gt;A,B,C,P,&lt;/math&gt; and &lt;math&gt;Q&lt;/math&gt; is &lt;math&gt;d&lt;/math&gt;. Find &lt;math&gt;d&lt;/math&gt;.<br /> <br /> [[2016 AIME II Problems/Problem 14 | Solution]]<br /> ==Problem 15==<br /> For &lt;math&gt;1 \leq i \leq 215&lt;/math&gt; let &lt;math&gt;a_i = \dfrac{1}{2^{i}}&lt;/math&gt; and &lt;math&gt;a_{216} = \dfrac{1}{2^{215}}&lt;/math&gt;. Let &lt;math&gt;x_1, x_2, ..., x_{216}&lt;/math&gt; be positive real numbers such that &lt;math&gt;\sum_{i=1}^{216} x_i=1&lt;/math&gt; and &lt;math&gt;\sum_{1 \leq i &lt; j \leq 216} x_ix_j = \dfrac{107}{215} + \sum_{i=1}^{216} \dfrac{a_i x_i^{2}}{2(1-a_i)}&lt;/math&gt;. The maximum possible value of &lt;math&gt;x_2=\dfrac{m}{n}&lt;/math&gt;, where &lt;math&gt;m&lt;/math&gt; and &lt;math&gt;n&lt;/math&gt; are relatively prime positive integers. Find &lt;math&gt;m+n&lt;/math&gt;.<br /> <br /> [[2016 AIME II Problems/Problem 15 | Solution]]<br /> <br /> <br /> {{AIME box|year=2016|n=II|before=[[2016 AIME I Problems]]|after=[[2017 AIME I Problems]]}}<br /> {{MAA Notice}}</div> Cosmicgenius https://artofproblemsolving.com/wiki/index.php?title=2016_AIME_II_Problems&diff=148431 2016 AIME II Problems 2021-03-04T01:11:00Z <p>Cosmicgenius: /* Problem 3 */</p> <hr /> <div>{{AIME Problems|year=2016|n=II}}<br /> ==Problem 1==<br /> Initially Alex, Betty, and Charlie had a total of &lt;math&gt;444&lt;/math&gt; peanuts. Charlie had the most peanuts, and Alex had the least. The three numbers of peanuts that each person had formed a geometric progression. Alex eats &lt;math&gt;5&lt;/math&gt; of his peanuts, Betty eats &lt;math&gt;9&lt;/math&gt; of her peanuts, and Charlie eats &lt;math&gt;25&lt;/math&gt; of his peanuts. Now the three numbers of peanuts each person has forms an arithmetic progression. Find the number of peanuts Alex had initially.<br /> <br /> [[2016 AIME II Problems/Problem 1 | Solution]]<br /> <br /> ==Problem 2==<br /> There is a &lt;math&gt;40\%&lt;/math&gt; chance of rain on Saturday and a &lt;math&gt;30\%&lt;/math&gt; chance of rain on Sunday. However, it is twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday. The probability that it rains at least one day this weekend is &lt;math&gt;\frac{a}{b}&lt;/math&gt;, where &lt;math&gt;a&lt;/math&gt; and &lt;math&gt;b&lt;/math&gt; are relatively prime positive integers. Find &lt;math&gt;a+b&lt;/math&gt;.<br /> <br /> [[2016 AIME II Problems/Problem 2 | Solution]]<br /> <br /> ==Problem 3==<br /> Let &lt;math&gt;x,y,&lt;/math&gt; and &lt;math&gt;z&lt;/math&gt; be real numbers satisfying the system<br /> &lt;cmath&gt;<br /> \begin{align*}<br /> \log_2(xyz-3+\log_5 x)&amp;=5,\\<br /> \log_3(xyz-3+\log_5 y)&amp;=4,\\<br /> \log_4(xyz-3+\log_5 z)&amp;=4.<br /> \end{align*}<br /> &lt;/cmath&gt;<br /> Find the value of &lt;math&gt;|\log_5 x|+|\log_5 y|+|\log_5 z|&lt;/math&gt;.<br /> <br /> [[2016 AIME II Problems/Problem 3 | Solution]]<br /> <br /> ==Problem 4==<br /> An &lt;math&gt;a \times b \times c&lt;/math&gt; rectangular box is built from &lt;math&gt;a \cdot b \cdot c&lt;/math&gt; unit cubes. Each unit cube is colored red, green, or yellow. Each of the &lt;math&gt;a&lt;/math&gt; layers of size &lt;math&gt;1 \times b \times c&lt;/math&gt; parallel to the &lt;math&gt;(b \times c)&lt;/math&gt; faces of the box contains exactly &lt;math&gt;9&lt;/math&gt; red cubes, exactly &lt;math&gt;12&lt;/math&gt; green cubes, and some yellow cubes. Each of the &lt;math&gt;b&lt;/math&gt; layers of size &lt;math&gt;a \times 1 \times c&lt;/math&gt; parallel to the &lt;math&gt;(a \times c)&lt;/math&gt; faces of the box contains exactly &lt;math&gt;20&lt;/math&gt; green cubes, exactly &lt;math&gt;25&lt;/math&gt; yellow cubes, and some red cubes. Find the smallest possible volume of the box.<br /> <br /> [[2016 AIME II Problems/Problem 4 | Solution]]<br /> ==Problem 5==<br /> Triangle &lt;math&gt;ABC_0&lt;/math&gt; has a right angle at &lt;math&gt;C_0&lt;/math&gt;. Its side lengths are pairwise relatively prime positive integers, and its perimeter is &lt;math&gt;p&lt;/math&gt;. Let &lt;math&gt;C_1&lt;/math&gt; be the foot of the altitude to &lt;math&gt;\overline{AB}&lt;/math&gt;, and for &lt;math&gt;n \geq 2&lt;/math&gt;, let &lt;math&gt;C_n&lt;/math&gt; be the foot of the altitude to &lt;math&gt;\overline{C_{n-2}B}&lt;/math&gt; in &lt;math&gt;\triangle C_{n-2}C_{n-1}B&lt;/math&gt;. The sum &lt;math&gt;\sum_{n=2}^\infty C_{n-2}C_{n-1} = 6p&lt;/math&gt;. Find &lt;math&gt;p&lt;/math&gt;.<br /> <br /> [[2016 AIME II Problems/Problem 5 | Solution]]<br /> <br /> ==Problem 6==<br /> For polynomial &lt;math&gt;P(x)=1-\dfrac{1}{3}x+\dfrac{1}{6}x^{2}&lt;/math&gt;, define<br /> &lt;math&gt;Q(x)=P(x)P(x^{3})P(x^{5})P(x^{7})P(x^{9})=\sum_{i=0}^{50} a_ix^{i}&lt;/math&gt;.<br /> Then &lt;math&gt;\sum_{i=0}^{50} |a_i|=\dfrac{m}{n}&lt;/math&gt;, where &lt;math&gt;m&lt;/math&gt; and &lt;math&gt;n&lt;/math&gt; are relatively prime positive integers. Find &lt;math&gt;m+n&lt;/math&gt;.<br /> <br /> [[2016 AIME II Problems/Problem 6 | Solution]]<br /> ==Problem 7==<br /> Squares &lt;math&gt;ABCD&lt;/math&gt; and &lt;math&gt;EFGH&lt;/math&gt; have a common center and &lt;math&gt;\overline{AB} || \overline{EF}&lt;/math&gt;. The area of &lt;math&gt;ABCD&lt;/math&gt; is 2016, and the area of &lt;math&gt;EFGH&lt;/math&gt; is a smaller positive integer. Square &lt;math&gt;IJKL&lt;/math&gt; is constructed so that each of its vertices lies on a side of &lt;math&gt;ABCD&lt;/math&gt; and each vertex of &lt;math&gt;EFGH&lt;/math&gt; lies on a side of &lt;math&gt;IJKL&lt;/math&gt;. Find the difference between the largest and smallest positive integer values for the area of &lt;math&gt;IJKL&lt;/math&gt;.<br /> <br /> [[2016 AIME II Problems/Problem 7 | Solution]]<br /> <br /> ==Problem 8==<br /> Find the number of sets &lt;math&gt;\{a,b,c\}&lt;/math&gt; of three distinct positive integers with the property that the product of &lt;math&gt;a,b,&lt;/math&gt; and &lt;math&gt;c&lt;/math&gt; is equal to the product of &lt;math&gt;11,21,31,41,51,&lt;/math&gt; and &lt;math&gt;61&lt;/math&gt;.<br /> <br /> [[2016 AIME II Problems/Problem 8 | Solution]]<br /> <br /> ==Problem 9==<br /> The sequences of positive integers &lt;math&gt;1,a_2, a_3,...&lt;/math&gt; and &lt;math&gt;1,b_2, b_3,...&lt;/math&gt; are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let &lt;math&gt;c_n=a_n+b_n&lt;/math&gt;. There is an integer &lt;math&gt;k&lt;/math&gt; such that &lt;math&gt;c_{k-1}=100&lt;/math&gt; and &lt;math&gt;c_{k+1}=1000&lt;/math&gt;. Find &lt;math&gt;c_k&lt;/math&gt;.<br /> <br /> [[2016 AIME II Problems/Problem 9 | Solution]]<br /> ==Problem 10==<br /> Triangle &lt;math&gt;ABC&lt;/math&gt; is inscribed in circle &lt;math&gt;\omega&lt;/math&gt;. Points &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;Q&lt;/math&gt; are on side &lt;math&gt;\overline{AB}&lt;/math&gt; with &lt;math&gt;AP&lt;AQ&lt;/math&gt;. Rays &lt;math&gt;CP&lt;/math&gt; and &lt;math&gt;CQ&lt;/math&gt; meet &lt;math&gt;\omega&lt;/math&gt; again at &lt;math&gt;S&lt;/math&gt; and &lt;math&gt;T&lt;/math&gt; (other than &lt;math&gt;C&lt;/math&gt;), respectively. If &lt;math&gt;AP=4,PQ=3,QB=6,BT=5,&lt;/math&gt; and &lt;math&gt;AS=7&lt;/math&gt;, then &lt;math&gt;ST=\frac{m}{n}&lt;/math&gt;, where &lt;math&gt;m&lt;/math&gt; and &lt;math&gt;n&lt;/math&gt; are relatively prime positive integers. Find &lt;math&gt;m+n&lt;/math&gt;.<br /> <br /> [[2016 AIME II Problems/Problem 10 | Solution]]<br /> ==Problem 11==<br /> For positive integers &lt;math&gt;N&lt;/math&gt; and &lt;math&gt;k&lt;/math&gt;, define &lt;math&gt;N&lt;/math&gt; to be &lt;math&gt;k&lt;/math&gt;-nice if there exists a positive integer &lt;math&gt;a&lt;/math&gt; such that &lt;math&gt;a^{k}&lt;/math&gt; has exactly &lt;math&gt;N&lt;/math&gt; positive divisors. Find the number of positive integers less than &lt;math&gt;1000&lt;/math&gt; that are neither &lt;math&gt;7&lt;/math&gt;-nice nor &lt;math&gt;8&lt;/math&gt;-nice.<br /> <br /> [[2016 AIME II Problems/Problem 11 | Solution]]<br /> ==Problem 12==<br /> The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.<br /> <br /> &lt;asy&gt;<br /> draw(Circle((0,0), 4));<br /> draw(Circle((0,0), 3));<br /> draw((0,4)--(0,3));<br /> draw((0,-4)--(0,-3));<br /> draw((-2.598, 1.5)--(-3.4641, 2));<br /> draw((-2.598, -1.5)--(-3.4641, -2));<br /> draw((2.598, -1.5)--(3.4641, -2));<br /> draw((2.598, 1.5)--(3.4641, 2));<br /> &lt;/asy&gt;<br /> <br /> [[2016 AIME II Problems/Problem 12 | Solution]]<br /> <br /> ==Problem 13==<br /> Beatrix is going to place six rooks on a &lt;math&gt;6 \times 6&lt;/math&gt; chessboard where both the rows and columns are labeled &lt;math&gt;1&lt;/math&gt; to &lt;math&gt;6&lt;/math&gt;; the rooks are placed so that no two rooks are in the same row or the same column. The &lt;math&gt;value&lt;/math&gt; of a square is the sum of its row number and column number. The &lt;math&gt;score&lt;/math&gt; of an arrangement of rooks is the least value of any occupied square.The average score over all valid configurations is &lt;math&gt;\frac{p}{q}&lt;/math&gt;, where &lt;math&gt;p&lt;/math&gt; and &lt;math&gt;q&lt;/math&gt; are relatively prime positive integers. Find &lt;math&gt;p+q&lt;/math&gt;.<br /> <br /> [[2016 AIME II Problems/Problem 13 | Solution]]<br /> <br /> ==Problem 14==<br /> Equilateral &lt;math&gt;\triangle ABC&lt;/math&gt; has side length &lt;math&gt;600&lt;/math&gt;. Points &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;Q&lt;/math&gt; lie outside the plane of &lt;math&gt;\triangle ABC&lt;/math&gt; and are on opposite sides of the plane. Furthermore, &lt;math&gt;PA=PB=PC&lt;/math&gt;, and &lt;math&gt;QA=QB=QC&lt;/math&gt;, and the planes of &lt;math&gt;\triangle PAB&lt;/math&gt; and &lt;math&gt;\triangle QAB&lt;/math&gt; form a &lt;math&gt;120^{\circ}&lt;/math&gt; dihedral angle (the angle between the two planes). There is a point &lt;math&gt;O&lt;/math&gt; whose distance from each of &lt;math&gt;A,B,C,P,&lt;/math&gt; and &lt;math&gt;Q&lt;/math&gt; is &lt;math&gt;d&lt;/math&gt;. Find &lt;math&gt;d&lt;/math&gt;.<br /> <br /> [[2016 AIME II Problems/Problem 14 | Solution]]<br /> ==Problem 15==<br /> For &lt;math&gt;1 \leq i \leq 215&lt;/math&gt; let &lt;math&gt;a_i = \dfrac{1}{2^{i}}&lt;/math&gt; and &lt;math&gt;a_{216} = \dfrac{1}{2^{215}}&lt;/math&gt;. Let &lt;math&gt;x_1, x_2, ..., x_{216}&lt;/math&gt; be positive real numbers such that &lt;math&gt;\sum_{i=1}^{216} x_i=1&lt;/math&gt; and &lt;math&gt;\sum_{1 \leq i &lt; j \leq 216} x_ix_j = \dfrac{107}{215} + \sum_{i=1}^{216} \dfrac{a_i x_i^{2}}{2(1-a_i)}&lt;/math&gt;. The maximum possible value of &lt;math&gt;x_2=\dfrac{m}{n}&lt;/math&gt;, where &lt;math&gt;m&lt;/math&gt; and &lt;math&gt;n&lt;/math&gt; are relatively prime positive integers. Find &lt;math&gt;m+n&lt;/math&gt;.<br /> <br /> [[2016 AIME II Problems/Problem 15 | Solution]]<br /> <br /> <br /> {{AIME box|year=2016|n=II|before=[[2016 AIME I Problems]]|after=[[2017 AIME I Problems]]}}<br /> {{MAA Notice}}</div> Cosmicgenius https://artofproblemsolving.com/wiki/index.php?title=2012_AMC_10A_Problems&diff=100765 2012 AMC 10A Problems 2019-01-22T01:15:54Z <p>Cosmicgenius: /* Problem 21 */</p> <hr /> <div>== Problem 1 ==<br /> <br /> Cagney can frost a cupcake every 20 seconds and Lacey can frost a cupcake every 30 seconds. Working together, how many cupcakes can they frost in 5 minutes?<br /> <br /> &lt;math&gt; \textbf{(A)}\ 10\qquad\textbf{(B)}\ 15\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 25\qquad\textbf{(E)}\ 30 &lt;/math&gt;<br /> <br /> [[2012 AMC 10A Problems/Problem 1|Solution]]<br /> <br /> == Problem 2 ==<br /> <br /> A square with side length 8 is cut in half, creating two congruent rectangles. What are the dimensions of one of these rectangles?<br /> <br /> &lt;math&gt; \textbf{(A)}\ 2\ \text{by}\ 4\qquad\textbf{(B)}\ \ 2\ \text{by}\ 6\qquad\textbf{(C)}\ \ 2\ \text{by}\ 8\qquad\textbf{(D)}\ 4\ \text{by}\ 4\qquad\textbf{(E)}\ 4\ \text{by}\ 8 &lt;/math&gt;<br /> <br /> [[2012 AMC 10A Problems/Problem 2|Solution]]<br /> <br /> == Problem 3 ==<br /> <br /> A bug crawls along a number line, starting at -2. It crawls to -6, then turns around and crawls to 5. How many units does the bug crawl altogether?<br /> <br /> &lt;math&gt; \textbf{(A)}\ 9\qquad\textbf{(B)}\ 11\qquad\textbf{(C)}\ 13\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 15 &lt;/math&gt;<br /> <br /> [[2012 AMC 10A Problems/Problem 3|Solution]]<br /> <br /> == Problem 4 ==<br /> <br /> Let &lt;math&gt;\angle ABC = 24^\circ &lt;/math&gt; and &lt;math&gt;\angle ABD = 20^\circ &lt;/math&gt;. What is the smallest possible degree measure for angle CBD?<br /> <br /> &lt;math&gt; \textbf{(A)}\ 0\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 12 &lt;/math&gt;<br /> <br /> [[2012 AMC 10A Problems/Problem 4|Solution]]<br /> <br /> == Problem 5 ==<br /> <br /> Last year 100 adult cats, half of whom were female, were brought into the Smallville Animal Shelter. Half of the adult female cats were accompanied by a litter of kittens. The average number of kittens per litter was 4. What was the total number of cats and kittens received by the shelter last year?<br /> <br /> &lt;math&gt; \textbf{(A)}\ 150\qquad\textbf{(B)}\ 200\qquad\textbf{(C)}\ 250\qquad\textbf{(D)}\ 300\qquad\textbf{(E)}\ 400 &lt;/math&gt;<br /> <br /> [[2012 AMC 10A Problems/Problem 5|Solution]]<br /> <br /> == Problem 6 ==<br /> <br /> The product of two positive numbers is 9. The reciprocal of one of these numbers is 4 times the reciprocal of the other number. What is the sum of the two numbers?<br /> <br /> &lt;math&gt; \textbf{(A)}\ \frac{10}{3}\qquad\textbf{(B)}\ \frac{20}{3}\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ \frac{15}{2}\qquad\textbf{(E)}\ 8 &lt;/math&gt;<br /> <br /> [[2012 AMC 10A Problems/Problem 6|Solution]]<br /> <br /> == Problem 7 ==<br /> <br /> In a bag of marbles, &lt;math&gt;\frac{3}{5}&lt;/math&gt; of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red?<br /> <br /> &lt;math&gt; \textbf{(A)}\ \frac{2}{5}\qquad\textbf{(B)}\ \frac{3}{7}\qquad\textbf{(C)}\ \frac{4}{7}\qquad\textbf{(D)}\ \frac{3}{5}\qquad\textbf{(E)}\ \frac{4}{5} &lt;/math&gt;<br /> <br /> [[2012 AMC 10A Problems/Problem 7|Solution]]<br /> <br /> == Problem 8 ==<br /> <br /> The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number?<br /> <br /> &lt;math&gt; \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 8 &lt;/math&gt;<br /> <br /> [[2012 AMC 10A Problems/Problem 8|Solution]]<br /> <br /> == Problem 9 ==<br /> <br /> A pair of six-sided dice are labeled so that one die has only even numbers (two each of 2, 4, and 6), and the other die has only odd numbers (two of each 1, 3, and 5). The pair of dice is rolled. What is the probability that the sum of the numbers on the tops of the two dice is 7?<br /> <br /> &lt;math&gt; \textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{1}{5}\qquad\textbf{(C)}\ \frac{1}{4}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{1}{2} &lt;/math&gt;<br /> <br /> [[2012 AMC 10A Problems/Problem 9|Solution]]<br /> <br /> == Problem 10 ==<br /> <br /> Mary divides a circle into 12 sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?<br /> <br /> &lt;math&gt; \textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 12 &lt;/math&gt;<br /> <br /> [[2012 AMC 10A Problems/Problem 10|Solution]]<br /> <br /> == Problem 11 ==<br /> <br /> Externally tangent circles with centers at points A and B have radii of lengths 5 and 3, respectively. A line externally tangent to both circles intersects ray AB at point C. What is BC?<br /> <br /> &lt;math&gt; \textbf{(A)}\ 4\qquad\textbf{(B)}\ 4.8\qquad\textbf{(C)}\ 10.2\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 14.4 &lt;/math&gt;<br /> <br /> [[2012 AMC 10A Problems/Problem 11|Solution]]<br /> <br /> == Problem 12 ==<br /> <br /> A year is a leap year if and only if the year number is divisible by 400 (such as 2000) or is divisible by 4 but not 100 (such as 2012). The 200th anniversary of the birth of novelist Charles Dickens was celebrated on February 7, 2012, a Tuesday. On what day of the week was Dickens born?<br /> <br /> &lt;math&gt; \textbf{(A)}\ \text{Friday}\qquad\textbf{(B)}\ \text{Saturday}\qquad\textbf{(C)}\ \text{Sunday}\qquad\textbf{(D)}\ \text{Monday}\qquad\textbf{(E)}\ \text{Tuesday} &lt;/math&gt;<br /> <br /> [[2012 AMC 10A Problems/Problem 12|Solution]]<br /> <br /> == Problem 13 ==<br /> <br /> An ''iterative average'' of the numbers 1, 2, 3, 4, and 5 is computed the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is the difference between the largest and smallest possible values that can be obtained using this procedure?<br /> <br /> &lt;math&gt; \textbf{(A)}\ \frac{31}{16}\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \frac{17}{8}\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ \frac{65}{16} &lt;/math&gt;<br /> <br /> [[2012 AMC 10A Problems/Problem 13|Solution]]<br /> <br /> == Problem 14 ==<br /> <br /> Chubby makes nonstandard checkerboards that have &lt;math&gt;31&lt;/math&gt; squares on each side. The checkerboards have a black square in every corner and alternate red and black squares along every row and column. How many black squares are there on such a checkerboard?<br /> <br /> &lt;math&gt; \textbf{(A)}\ 480 \qquad\textbf{(B)}\ 481 \qquad\textbf{(C)}\ 482 \qquad\textbf{(D)}\ 483 \qquad\textbf{(E)}\ 484&lt;/math&gt;<br /> <br /> [[2012 AMC 10A Problems/Problem 14|Solution]]<br /> <br /> == Problem 15 ==<br /> <br /> Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of &lt;math&gt;\triangle ABC&lt;/math&gt;?<br /> <br /> &lt;center&gt;&lt;asy&gt;<br /> size(200);<br /> unitsize(2cm);<br /> defaultpen(linewidth(.8pt)+fontsize(10pt));<br /> dotfactor=4;<br /> <br /> pair A=(0,0), B=(1,0); pair C=(0.8,-0.4);<br /> draw(A--(2,0)); draw((0,-1)--(2,-1)); draw((0,-2)--(1,-2));<br /> draw(A--(0,-2)); draw(B--(1,-2)); draw((2,0)--(2,-1));<br /> draw(A--(2,-1)); draw(B--(0,-2));<br /> <br /> pair[] ps={A,B,C};<br /> dot(ps);<br /> <br /> label(&quot;$A$&quot;,A,N);<br /> label(&quot;$B$&quot;,B,N);<br /> label(&quot;$C$&quot;,C,W);<br /> &lt;/asy&gt;&lt;/center&gt;<br /> <br /> &lt;math&gt; \textbf{(A)}\ \frac16 \qquad\textbf{(B)}\ \frac15 \qquad\textbf{(C)}\ \frac29 \qquad\textbf{(D)}\ \frac13 \qquad\textbf{(E)}\ \frac{\sqrt{2}}{4}&lt;/math&gt;<br /> <br /> [[2012 AMC 10A Problems/Problem 15|Solution]]<br /> <br /> == Problem 16 ==<br /> <br /> Three runners start running simultaneously from the same point on a 500-meter circular track. They each run clockwise around the course maintaining constant speeds of 4.4, 4.8, and 5.0 meters per second. The runners stop once they are all together again somewhere on the circular course. How many seconds do the runners run?<br /> <br /> &lt;math&gt; \textbf{(A)}\ 1,000\qquad\textbf{(B)}\ 1,250\qquad\textbf{(C)}\ 2,500\qquad\textbf{(D)}\ 5,000\qquad\textbf{(E)}\ 10,000 &lt;/math&gt;<br /> <br /> [[2012 AMC 10A Problems/Problem 16|Solution]]<br /> <br /> == Problem 17 ==<br /> <br /> Let &lt;math&gt;a&lt;/math&gt; and &lt;math&gt;b&lt;/math&gt; be relatively prime integers with &lt;math&gt;a&gt;b&gt;0&lt;/math&gt; and &lt;math&gt;\frac{a^3-b^3}{(a-b)^3}&lt;/math&gt; = &lt;math&gt;\frac{73}{3}&lt;/math&gt;. What is &lt;math&gt;a-b&lt;/math&gt;?<br /> <br /> &lt;math&gt; \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 &lt;/math&gt;<br /> <br /> [[2012 AMC 10A Problems/Problem 17|Solution]]<br /> <br /> == Problem 18 ==<br /> <br /> The closed curve in the figure is made up of 9 congruent circular arcs each of length &lt;math&gt;\frac{2\pi}{3}&lt;/math&gt;, where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side 2. What is the area enclosed by the curve?<br /> <br /> &lt;asy&gt;<br /> size(5cm);<br /> defaultpen(fontsize(6pt));<br /> dotfactor=4;<br /> label(&quot;$\circ$&quot;,(0,1));<br /> label(&quot;$\circ$&quot;,(0.865,0.5));<br /> label(&quot;$\circ$&quot;,(-0.865,0.5));<br /> label(&quot;$\circ$&quot;,(0.865,-0.5));<br /> label(&quot;$\circ$&quot;,(-0.865,-0.5));<br /> label(&quot;$\circ$&quot;,(0,-1));<br /> dot((0,1.5));<br /> dot((-0.4325,0.75));<br /> dot((0.4325,0.75));<br /> dot((-0.4325,-0.75));<br /> dot((0.4325,-0.75));<br /> dot((-0.865,0));<br /> dot((0.865,0));<br /> dot((-1.2975,-0.75));<br /> dot((1.2975,-0.75));<br /> draw(Arc((0,1),0.5,210,-30));<br /> draw(Arc((0.865,0.5),0.5,150,270));<br /> draw(Arc((0.865,-0.5),0.5,90,-150));<br /> draw(Arc((0.865,-0.5),0.5,90,-150));<br /> draw(Arc((0,-1),0.5,30,150));<br /> draw(Arc((-0.865,-0.5),0.5,330,90));<br /> draw(Arc((-0.865,0.5),0.5,-90,30));<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A)}\ 2\pi+6\qquad\textbf{(B)}\ 2\pi+4\sqrt{3}\qquad\textbf{(C)}\ 3\pi+4\qquad\textbf{(D)}\ 2\pi+3\sqrt{3}+2\qquad\textbf{(E)}\ \pi+6\sqrt{3}&lt;/math&gt;<br /> <br /> [[2012 AMC 10A Problems/Problem 18|Solution]]<br /> <br /> == Problem 19 ==<br /> <br /> Paula the painter and her two helpers each paint at constant, but different, rates. They always start at 8:00 AM, and all three always take the same amount of time to eat lunch. On Monday the three of them painted 50% of a house, quitting at 4:00 PM. On Tuesday, when Paula wasn't there, the two helpers painted only 24% of the house and quit at 2:12 PM. On Wednesday Paula worked by herself and finished the house by working until 7:12 P.M. How long, in minutes, was each day's lunch break?<br /> <br /> &lt;math&gt; \textbf{(A)}\ 30\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 60 &lt;/math&gt;<br /> <br /> [[2012 AMC 10A Problems/Problem 19|Solution]]<br /> <br /> == Problem 20 ==<br /> <br /> A &lt;math&gt;3&lt;/math&gt; x &lt;math&gt;3&lt;/math&gt; square is partitioned into &lt;math&gt;9&lt;/math&gt; unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated &lt;math&gt;90\,^{\circ}&lt;/math&gt; clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability the grid is now entirely black?<br /> <br /> &lt;math&gt; \textbf{(A)}\ \frac{49}{512}\qquad\textbf{(B)}\ \frac{7}{64}\qquad\textbf{(C)}\ \frac{121}{1024}\qquad\textbf{(D)}\ \frac{81}{512}\qquad\textbf{(E)}\ \frac{9}{32} &lt;/math&gt;<br /> <br /> [[2012 AMC 10A Problems/Problem 20|Solution]]<br /> <br /> == Problem 21 ==<br /> <br /> Let points &lt;math&gt;A = (0 ,0 ,0)&lt;/math&gt;, &lt;math&gt;B = (1, 0, 0)&lt;/math&gt;, &lt;math&gt;C = (0, 2, 0)&lt;/math&gt;, and &lt;math&gt;D = (0, 0, 3)&lt;/math&gt;. Points &lt;math&gt;E&lt;/math&gt;, &lt;math&gt;F&lt;/math&gt;, &lt;math&gt;G&lt;/math&gt;, and &lt;math&gt;H&lt;/math&gt; are midpoints of line segments &lt;math&gt;\overline{BD},\text{ } \overline{AB}, \text{ } \overline {AC},&lt;/math&gt; and &lt;math&gt;\overline{DC}&lt;/math&gt; respectively. What is the area of &lt;math&gt;EFGH&lt;/math&gt;?<br /> <br /> &lt;math&gt; \textbf{(A)}\ \sqrt{2}\qquad\textbf{(B)}\ \frac{2\sqrt{5}}{3}\qquad\textbf{(C)}\ \frac{3\sqrt{5}}{4}\qquad\textbf{(D)}\ \sqrt{3}\qquad\textbf{(E)}\ \frac{2\sqrt{7}}{3} &lt;/math&gt;<br /> <br /> [[2012 AMC 10A Problems/Problem 21|Solution]]<br /> <br /> == Problem 22 ==<br /> <br /> The sum of the first &lt;math&gt;m&lt;/math&gt; positive odd integers is 212 more than the sum of the first &lt;math&gt;n&lt;/math&gt; positive even integers. What is the sum of all possible values of &lt;math&gt;n&lt;/math&gt;?<br /> <br /> &lt;math&gt; \textbf{(A)}\ 255\qquad\textbf{(B)}\ 256\qquad\textbf{(C)}\ 257\qquad\textbf{(D)}\ 258\qquad\textbf{(E)}\ 259 &lt;/math&gt;<br /> <br /> [[2012 AMC 10A Problems/Problem 22|Solution]]<br /> <br /> == Problem 23 ==<br /> <br /> Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen?<br /> <br /> &lt;math&gt; \textbf{(A)}\ 60\qquad\textbf{(B)}\ 170\qquad\textbf{(C)}\ 290\qquad\textbf{(D)}\ 320\qquad\textbf{(E)}\ 660 &lt;/math&gt;<br /> <br /> [[2012 AMC 10A Problems/Problem 23|Solution]]<br /> <br /> == Problem 24 ==<br /> <br /> Let &lt;math&gt;a&lt;/math&gt;, &lt;math&gt;b&lt;/math&gt;, and &lt;math&gt;c&lt;/math&gt; be positive integers with &lt;math&gt;a\ge&lt;/math&gt; &lt;math&gt;b\ge&lt;/math&gt; &lt;math&gt;c&lt;/math&gt; such that<br /> &lt;cmath&gt;\begin{align*}a^2-b^2-c^2+ab&amp;=2011\text{ and}\\ a^2+3b^2+3c^2-3ab-2ac-2bc&amp;=-1997.\end{align*}&lt;/cmath&gt;<br /> <br /> What is &lt;math&gt;a&lt;/math&gt;?<br /> <br /> &lt;math&gt; \textbf{(A)}\ 249\qquad\textbf{(B)}\ 250\qquad\textbf{(C)}\ 251\qquad\textbf{(D)}\ 252\qquad\textbf{(E)}\ 253 &lt;/math&gt;<br /> <br /> [[2012 AMC 10A Problems/Problem 24|Solution]]<br /> <br /> == Problem 25 ==<br /> <br /> Real numbers &lt;math&gt;x&lt;/math&gt;, &lt;math&gt;y&lt;/math&gt;, and &lt;math&gt;z&lt;/math&gt; are chosen independently and at random from the interval &lt;math&gt;[0,n]&lt;/math&gt; for some positive integer &lt;math&gt;n&lt;/math&gt;. The probability that no two of &lt;math&gt;x&lt;/math&gt;, &lt;math&gt;y&lt;/math&gt;, and &lt;math&gt;z&lt;/math&gt; are within 1 unit of each other is greater than &lt;math&gt;\frac {1}{2}&lt;/math&gt;. What is the smallest possible value of &lt;math&gt;n&lt;/math&gt;?<br /> <br /> &lt;math&gt; \textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11 &lt;/math&gt;<br /> <br /> [[2012 AMC 10A Problems/Problem 25|Solution]]<br /> ==See also==<br /> {{AMC10 box|year=2012|ab=A|before=[[2011 AMC 10B Problems]]|after=[[2012 AMC 10B Problems]]}}<br /> * [[AMC 10]]<br /> * [[AMC 10 Problems and Solutions]]<br /> * [[2012 AMC 12A]]<br /> * [[Mathematics competition resources]]<br /> {{MAA Notice}}</div> Cosmicgenius https://artofproblemsolving.com/wiki/index.php?title=2000_AMC_12_Problems/Problem_24&diff=99417 2000 AMC 12 Problems/Problem 24 2018-12-13T04:15:11Z <p>Cosmicgenius: /* Problem */</p> <hr /> <div>== Problem ==<br /> [[Image:2000_12_AMC-24.png|right]]<br /> If circular [[arc]]s &lt;math&gt;AC&lt;/math&gt; and &lt;math&gt;BC&lt;/math&gt; have [[center]]s at &lt;math&gt;B&lt;/math&gt; and &lt;math&gt;A&lt;/math&gt;, respectively, then there exists a [[circle]] [[tangent (geometry)|tangent]] to both &lt;math&gt;\overarc{AC}&lt;/math&gt; and &lt;math&gt;\overarc{BC}&lt;/math&gt;, and to &lt;math&gt;\overline{AB}&lt;/math&gt;. If the length of &lt;math&gt;\overarc{BC}&lt;/math&gt; is &lt;math&gt;12&lt;/math&gt;, then the [[circumference]] of the circle is <br /> <br /> &lt;math&gt;\text {(A)}\ 24 \qquad \text {(B)}\ 25 \qquad \text {(C)}\ 26 \qquad \text {(D)}\ 27 \qquad \text {(E)}\ 28&lt;/math&gt;<br /> <br /> == Solution 1 ==<br /> [[Image:2000_12_AMC-24a.png|left]]<br /> <br /> Since &lt;math&gt;AB,BC,AC&lt;/math&gt; are all [[radius|radii]], it follows that &lt;math&gt;\triangle ABC&lt;/math&gt; is an [[equilateral triangle]]. <br /> <br /> Draw the circle with center &lt;math&gt;A&lt;/math&gt; and radius &lt;math&gt;\overline{AB}&lt;/math&gt;. Then let &lt;math&gt;D&lt;/math&gt; be the point of tangency of the two circles, and &lt;math&gt;E&lt;/math&gt; be the intersection of the smaller circle and &lt;math&gt;\overline{AD}&lt;/math&gt;. Let &lt;math&gt;F&lt;/math&gt; be the intersection of the smaller circle and &lt;math&gt;\overline{AB}&lt;/math&gt;. Also define the radii &lt;math&gt;r_1 = AB, r_2 = \frac{DE}{2}&lt;/math&gt; (note that &lt;math&gt;DE&lt;/math&gt; is a diameter of the smaller circle, as &lt;math&gt;D&lt;/math&gt; is the point of tangency of both circles, the radii of a circle is perpendicular to the tangent, hence the two centers of the circle are collinear with each other and &lt;math&gt;D&lt;/math&gt;).<br /> <br /> By the [[Power of a Point Theorem]], <br /> &lt;cmath&gt;AF^2 = AE \cdot AD \Longrightarrow \left(\frac {r_1}2\right)^2 = (AD - 2r_2) \cdot AD.&lt;/cmath&gt;<br /> <br /> Since &lt;math&gt;AD = r_1&lt;/math&gt;, then &lt;math&gt;\frac{r_1^2}{4} = r_1 (r_1 - 2r_2) \Longrightarrow r_2 = \frac{3r_1}{8}&lt;/math&gt;. Since &lt;math&gt;ABC&lt;/math&gt; is equilateral, &lt;math&gt;\angle BAC = 60^{\circ}&lt;/math&gt;, and so &lt;math&gt;\stackrel{\frown}{BC} = 12 = \frac{60}{360} 2\pi r_1 \Longrightarrow r_1 = \frac{36}{\pi}&lt;/math&gt;. Thus &lt;math&gt;r_2 = \frac{27}{2\pi}&lt;/math&gt; and the circumference of the circle is &lt;math&gt;27\ \mathrm{(D)}&lt;/math&gt;.<br /> <br /> (Alternatively, the [[Pythagorean Theorem]] can also be used to find &lt;math&gt;r_2&lt;/math&gt; in terms of &lt;math&gt;r_1&lt;/math&gt;. Notice that since AB is tangent to circle &lt;math&gt;O&lt;/math&gt;, &lt;math&gt;\overline{OF}&lt;/math&gt; is perpendicular to &lt;math&gt;\overline{AF}&lt;/math&gt;. Therefore, <br /> <br /> &lt;cmath&gt;AF^2 + OF^2 = AO^2&lt;/cmath&gt;<br /> &lt;cmath&gt;\left(\frac {r_1}{2}\right)^2 + r_2^2 = (r_1 - r_2)^2&lt;/cmath&gt;<br /> <br /> After simplification, &lt;math&gt;r_2 = \frac{3r_1}{8}&lt;/math&gt;.<br /> <br /> == Solution 2 (Pythagorean Theorem) ==<br /> First, note the triangle &lt;math&gt;ABC&lt;/math&gt; is equilateral. Next, notice that since the arc &lt;math&gt;BC&lt;/math&gt; has length 12, it follows that we can find the radius of the sector centered at &lt;math&gt;A&lt;/math&gt;. &lt;math&gt;\frac {1}{6}({2}{\pi})AB=12 \implies AB=36/{\pi}&lt;/math&gt;. Next, connect the center of the circle to side &lt;math&gt;AB&lt;/math&gt;, and call this length &lt;math&gt;r&lt;/math&gt;, and call the foot &lt;math&gt;M&lt;/math&gt;. Since &lt;math&gt;ABC&lt;/math&gt; is equilateral, it follows that &lt;math&gt;MB=18/{\pi}&lt;/math&gt;, and &lt;math&gt;OA&lt;/math&gt; (where O is the center of the circle) is &lt;math&gt;36/{\pi}-r&lt;/math&gt;. By the pythagorean theorem, you get &lt;math&gt;r^2+(18/{\pi})^2=(36/{\pi}-r)^2 \implies r=27/2{\pi}&lt;/math&gt;. Finally, we see that the circumference is &lt;math&gt;2{\pi}*27/2{\pi}=\boxed{(D)27}&lt;/math&gt;.<br /> <br /> == See also ==<br /> {{AMC12 box|year=2000|num-b=23|num-a=25}}<br /> <br /> [[Category:Introductory Geometry Problems]]<br /> {{MAA Notice}}</div> Cosmicgenius https://artofproblemsolving.com/wiki/index.php?title=2018_AMC_8_Problems&diff=98744 2018 AMC 8 Problems 2018-11-21T16:54:29Z <p>Cosmicgenius: /* Problem 23 */</p> <hr /> <div>==Problem 1==<br /> An amusement park has a collection of scale models, with ratio &lt;math&gt;1 : 20&lt;/math&gt;, of buildings and other sights from around the country. The height of the United States Capitol is 289 feet. What is the height in feet of its replica to the nearest whole number?<br /> <br /> &lt;math&gt;\textbf{(A) }14\qquad\textbf{(B) }15\qquad\textbf{(C) }16\qquad\textbf{(D) }18\qquad\textbf{(E) }20&lt;/math&gt;<br /> <br /> [[2018 AMC 8 Problems/Problem 1|Solution]]<br /> <br /> ==Problem 2==<br /> What is the value of the product&lt;cmath&gt;\left(1+\frac{1}{1}\right)\cdot\left(1+\frac{1}{2}\right)\cdot\left(1+\frac{1}{3}\right)\cdot\left(1+\frac{1}{4}\right)\cdot\left(1+\frac{1}{5}\right)\cdot\left(1+\frac{1}{6}\right)?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{7}{6}\qquad\textbf{(B) }\frac{4}{3}\qquad\textbf{(C) }\frac{7}{2}\qquad\textbf{(D) }7\qquad\textbf{(E) }8&lt;/math&gt;<br /> <br /> [[2018 AMC 8 Problems/Problem 2|Solution]]<br /> ==Problem 3==<br /> Students Arn, Bob, Cyd, Dan, Eve, and Fon are arranged in that order in a circle. They start counting: Arn first, then Bob, and so forth. When the number contains a 7 as a digit (such as 47) or is a multiple of 7 that person leaves the circle and the counting continues. Who is the last one present in the circle?<br /> <br /> &lt;math&gt;\textbf{(A) } \text{Arn}\qquad\textbf{(B) }\text{Bob}\qquad\textbf{(C) }\text{Cyd}\qquad\textbf{(D) }\text{Dan}\qquad \textbf{(E) }\text{Eve}&lt;/math&gt;<br /> <br /> [[2018 AMC 8 Problems/Problem 3|Solution]]<br /> ==Problem 4==<br /> The twelve-sided figure shown has been drawn on &lt;math&gt;1 \text{ cm}\times 1 \text{ cm}&lt;/math&gt; graph paper. What is the area of the figure in &lt;math&gt;\text{cm}^2&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> unitsize(8mm);<br /> for (int i=0; i&lt;7; ++i) {<br /> draw((i,0)--(i,7),gray);<br /> draw((0,i+1)--(7,i+1),gray);<br /> }<br /> draw((1,3)--(2,4)--(2,5)--(3,6)--(4,5)--(5,5)--(6,4)--(5,3)--(5,2)--(4,1)--(3,2)--(2,2)--cycle,black+2bp);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) } 12 \qquad \textbf{(B) } 12.5 \qquad \textbf{(C) } 13 \qquad \textbf{(D) } 13.5 \qquad \textbf{(E) } 14&lt;/math&gt;<br /> <br /> [[2018 AMC 8 Problems/Problem 4|Solution]]<br /> ==Problem 5==<br /> What is the value of &lt;math&gt;1+3+5+\cdots+2017+2019-2-4-6-\cdots-2016-2018&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }-1010\qquad\textbf{(B) }-1009\qquad\textbf{(C) }1008\qquad\textbf{(D) }1009\qquad \textbf{(E) }1010&lt;/math&gt;<br /> <br /> [[2018 AMC 8 Problems/Problem 5|Solution]]<br /> ==Problem 6==<br /> On a trip to the beach, Anh traveled 50 miles on the highway and 10 miles on a coastal access road. He drove three times as fast on the highway as on the coastal road. If Anh spent 30 minutes driving on the coastal road, how many minutes did his entire trip take?<br /> <br /> &lt;math&gt;\textbf{(A) }50\qquad\textbf{(B) }70\qquad\textbf{(C) }80\qquad\textbf{(D) }90\qquad \textbf{(E) }100&lt;/math&gt;<br /> <br /> [[2018 AMC 8 Problems/Problem 6|Solution]]<br /> ==Problem 7==<br /> The &lt;math&gt;5&lt;/math&gt;-digit number &lt;math&gt;\underline{2}&lt;/math&gt; &lt;math&gt;\underline{0}&lt;/math&gt; &lt;math&gt;\underline{1}&lt;/math&gt; &lt;math&gt;\underline{8}&lt;/math&gt; &lt;math&gt;\underline{U}&lt;/math&gt; is divisible by &lt;math&gt;9&lt;/math&gt;. What is the remainder when this number is divided by &lt;math&gt;8&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }1\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }7&lt;/math&gt;<br /> <br /> [[2018 AMC 7 Problems/Problem 1|Solution]]<br /> ==Problem 8==<br /> Mr. Garcia asked the members of his health class how many days last week they exercised for at least 30 minutes. The results are summarized in the following bar graph, where the heights of the bars represent the number of students.<br /> <br /> &lt;asy&gt;<br /> size(8cm);<br /> void drawbar(real x, real h) {<br /> fill((x-0.15,0.5)--(x+0.15,0.5)--(x+0.15,h)--(x-0.15,h)--cycle,gray);<br /> }<br /> draw((0.5,0.5)--(7.5,0.5)--(7.5,5)--(0.5,5)--cycle);<br /> for (real i=1; i&lt;5; i=i+0.5) {<br /> draw((0.5,i)--(7.5,i),gray);<br /> }<br /> drawbar(1.0,1.0);<br /> drawbar(2.0,2.0);<br /> drawbar(3.0,1.5);<br /> drawbar(4.0,3.5);<br /> drawbar(5.0,4.5);<br /> drawbar(6.0,2.0);<br /> drawbar(7.0,1.5);<br /> for (int i=1; i&lt;8; ++i) {<br /> label(&quot;$&quot;+string(i)+&quot;$&quot;,(i,0.25));<br /> }<br /> for (int i=1; i&lt;9; ++i) {<br /> label(&quot;$&quot;+string(i)+&quot;$&quot;,(0.5,0.5*(i+1)),W);<br /> }<br /> label(&quot;Number of Days of Exercise&quot;,(4,-0.1));<br /> label(rotate(90)*&quot;Number of Students&quot;,(-0.1,2.75));<br /> &lt;/asy&gt;<br /> What was the mean number of days of exercise last week, rounded to the nearest hundredth, reported by the students in Mr. Garcia's class?<br /> <br /> &lt;math&gt;\textbf{(A) } 3.50 \qquad \textbf{(B) } 3.57 \qquad \textbf{(C) } 4.36 \qquad \textbf{(D) } 4.50 \qquad \textbf{(E) } 5.00&lt;/math&gt;<br /> <br /> [[2018 AMC 8 Problems/Problem 8|Solution]]<br /> ==Problem 9==<br /> Tyler is tiling the floor of his 12 foot by 16 foot living room. He plans to place one-foot by one-foot square tiles to form a border along the edges of the room and to fill in the rest of the floor with two-foot by two-foot square tiles. How many tiles will he use?<br /> <br /> &lt;math&gt;\textbf{(A) }48\qquad\textbf{(B) }87\qquad\textbf{(C) }91\qquad\textbf{(D) }96\qquad \textbf{(E) }120&lt;/math&gt;<br /> <br /> [[2018 AMC 8 Problems/Problem 10|Solution]]<br /> ==Problem 10==<br /> The [i]]harmonic mean[/i]] of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{3}{7}\qquad\textbf{(B) }\frac{7}{12}\qquad\textbf{(C) }\frac{12}{7}\qquad\textbf{(D) }\frac{7}{4}\qquad \textbf{(E) }\frac{7}{3}&lt;/math&gt;<br /> ==Problem 11==<br /> Abby, Bridget, and four of their classmates will be seated in two rows of three for a group picture, as shown.<br /> \begin{eqnarray*}<br /> \text{X}&amp;\quad\text{X}\quad&amp;\text{X} \\<br /> \text{X}&amp;\quad\text{X}\quad&amp;\text{X} <br /> \end{eqnarray*}<br /> If the seating positions are assigned randomly, what is the probability that Abby and Bridget are adjacent to each other in the same row or the same column?<br /> <br /> &lt;math&gt;\textbf{(A) } \frac{1}{3} \qquad \textbf{(B) } \frac{2}{5} \qquad \textbf{(C) } \frac{7}{15} \qquad \textbf{(D) } \frac{1}{2} \qquad \textbf{(E) } \frac{2}{3}&lt;/math&gt;<br /> <br /> [[2018 AMC 8 Problems/Problem 11|Solution]]<br /> ==Problem 12==<br /> The clock in Sri's car, which is not accurate, gains time at a constant rate. One day as he begins shopping he notes that his car clock and his watch (which is accurate) both say 12:00 noon. When he is done shopping, his watch says 12:30 and his car clock says 12:35. Later that day, Sri loses his watch. He looks at his car clock and it says 7:00. What is the actual time?<br /> <br /> &lt;math&gt;\textbf{(A) }5:50\qquad\textbf{(B) }6:00\qquad\textbf{(C) }6:30\qquad\textbf{(D) }6:55\qquad \textbf{(E) }8:10&lt;/math&gt;<br /> <br /> [[2018 AMC 8 Problems/Problem 12|Solution]]<br /> ==Problem 13==<br /> Laila took five math tests, each worth a maximum of 100 points. Laila's score on each test was an integer between 0 and 100, inclusive. Laila received the same score on the first four tests, and she received a higher score on the last test. Her average score on the five tests was 82. How many values are possible for Laila's score on the last test?<br /> <br /> &lt;math&gt;\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad \textbf{(E) }18&lt;/math&gt;<br /> <br /> [[2018 AMC 8 Problems/Problem 13|Solution]]<br /> <br /> ==Problem 14==<br /> Let &lt;math&gt;N&lt;/math&gt; be the greatest five-digit number whose digits have a product of &lt;math&gt;120&lt;/math&gt;. What is the sum of the digits of &lt;math&gt;N&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }15\qquad\textbf{(B) }16\qquad\textbf{(C) }17\qquad\textbf{(D) }18\qquad\textbf{(E) }20&lt;/math&gt;<br /> <br /> [[2018 AMC 8 Problems/Problem 14|Solution]]<br /> ==Problem 15==<br /> In the diagram below, a diameter of each of the two smaller circles is a radius of the larger circle. If the two smaller circles have a combined area of &lt;math&gt;1&lt;/math&gt; square unit, then what is the area of the shaded region, in square units?<br /> <br /> &lt;asy&gt;<br /> size(4cm);<br /> filldraw(scale(2)*unitcircle,gray,black);<br /> filldraw(shift(-1,0)*unitcircle,white,black);<br /> filldraw(shift(1,0)*unitcircle,white,black);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) } \frac{1}{4} \qquad \textbf{(B) } \frac{1}{3} \qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) } 1 \qquad \textbf{(E) } \frac{\pi}{2}&lt;/math&gt;<br /> <br /> [[2018 AMC 8 Problems/Problem 15|Solution]]<br /> ==Problem 16==<br /> Professor Chang has nine different language books lined up on a bookshelf: two Arabic, three German, and four Spanish. How many ways are there to arrange the nine books on the shelf keeping the Arabic books together and keeping the Spanish books together?<br /> <br /> &lt;math&gt;\textbf{(A) }1440\qquad\textbf{(B) }2880\qquad\textbf{(C) }5760\qquad\textbf{(D) }182,440\qquad \textbf{(E) }362,880&lt;/math&gt;<br /> <br /> [[2018 AMC 8 Problems/Problem 16|Solution]]<br /> ==Problem 17==<br /> Bella begins to walk from her house toward her friend Ella's house. At the same time, Ella begins to ride her bicycle toward Bella's house. They each maintain a constant speed, and Ella rides 5 times as fast as Bella walks. The distance between their houses is &lt;math&gt;2&lt;/math&gt; miles, which is &lt;math&gt;10,560&lt;/math&gt; feet, and Bella covers &lt;math&gt;2 \tfrac{1}{2}&lt;/math&gt; feet with each step. How many steps will Bella take by the time she meets Ella?<br /> <br /> &lt;math&gt;\textbf{(A) }704\qquad\textbf{(B) }845\qquad\textbf{(C) }1056\qquad\textbf{(D) }1760\qquad \textbf{(E) }3520&lt;/math&gt;<br /> <br /> [[2018 AMC 8 Problems/Problem 17|Solution]]<br /> ==Problem 18==<br /> How many positive factors does &lt;math&gt;23,232&lt;/math&gt; have?<br /> <br /> &lt;math&gt;\textbf{(A) }9\qquad\textbf{(B) }12\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }42&lt;/math&gt;<br /> <br /> [[2018 AMC 8 Problems/Problem 18|Solution]]<br /> ==Problem 19==<br /> In a sign pyramid a cell gets a &quot;+&quot; if the two cells below it have the same sign, and it gets a &quot;-&quot; if the two cells below it have different signs. The diagram below illustrates a sign pyramid with four levels. How many possible ways are there to fill the four cells in the bottom row to produce a &quot;+&quot; at the top of the pyramid?<br /> <br /> &lt;asy&gt;<br /> unitsize(2cm);<br /> path box = (-0.5,-0.2)--(-0.5,0.2)--(0.5,0.2)--(0.5,-0.2)--cycle;<br /> draw(box); label(&quot;$+$&quot;,(0,0));<br /> draw(shift(1,0)*box); label(&quot;$-$&quot;,(1,0));<br /> draw(shift(2,0)*box); label(&quot;$+$&quot;,(2,0));<br /> draw(shift(3,0)*box); label(&quot;$-$&quot;,(3,0));<br /> draw(shift(0.5,0.4)*box); label(&quot;$-$&quot;,(0.5,0.4));<br /> draw(shift(1.5,0.4)*box); label(&quot;$-$&quot;,(1.5,0.4));<br /> draw(shift(2.5,0.4)*box); label(&quot;$-$&quot;,(2.5,0.4));<br /> draw(shift(1,0.8)*box); label(&quot;$+$&quot;,(1,0.8));<br /> draw(shift(2,0.8)*box); label(&quot;$+$&quot;,(2,0.8));<br /> draw(shift(1.5,1.2)*box); label(&quot;$+$&quot;,(1.5,1.2));<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) } 2 \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 12 \qquad \textbf{(E) } 16&lt;/math&gt;<br /> <br /> [[2018 AMC 8 Problems/Problem 19|Solution]]<br /> ==Problem 20==<br /> In &lt;math&gt;\triangle ABC,&lt;/math&gt; a point &lt;math&gt;E&lt;/math&gt; is on &lt;math&gt;\overline{AB}&lt;/math&gt; with &lt;math&gt;AE=1&lt;/math&gt; and &lt;math&gt;EB=2.&lt;/math&gt; Point &lt;math&gt;D&lt;/math&gt; is on &lt;math&gt;\overline{AC}&lt;/math&gt; so that &lt;math&gt;\overline{DE} \parallel \overline{BC}&lt;/math&gt; and point &lt;math&gt;F&lt;/math&gt; is on &lt;math&gt;\overline{BC}&lt;/math&gt; so that &lt;math&gt;\overline{EF} \parallel \overline{AC}.&lt;/math&gt; What is the ratio of the area of &lt;math&gt;CDEF&lt;/math&gt; to the area of &lt;math&gt;\triangle ABC?&lt;/math&gt;<br /> <br /> &lt;asy&gt;<br /> size(7cm);<br /> pair A,B,C,DD,EE,FF;<br /> A = (0,0); B = (3,0); C = (0.5,2.5);<br /> EE = (1,0);<br /> DD = intersectionpoint(A--C,EE--EE+(C-B));<br /> FF = intersectionpoint(B--C,EE--EE+(C-A));<br /> draw(A--B--C--A--DD--EE--FF,black+1bp);<br /> label(&quot;$A$&quot;,A,S); label(&quot;$B$&quot;,B,S); label(&quot;$C$&quot;,C,N);<br /> label(&quot;$D$&quot;,DD,W); label(&quot;$E$&quot;,EE,S); label(&quot;$F$&quot;,FF,NE);<br /> label(&quot;$1$&quot;,(A+EE)/2,S); label(&quot;$2$&quot;,(EE+B)/2,S);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) } \frac{4}{9} \qquad \textbf{(B) } \frac{1}{2} \qquad \textbf{(C) } \frac{5}{9} \qquad \textbf{(D) } \frac{3}{5} \qquad \textbf{(E) } \frac{2}{3}&lt;/math&gt;<br /> <br /> [[2018 AMC 8 Problems/Problem 20|Solution]]<br /> ==Problem 21==<br /> How many positive three-digit integers have a remainder of 2 when divided by 6, a remainder of 5 when divided by 9, and a remainder of 7 when divided by 11?<br /> <br /> &lt;math&gt;\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad \textbf{(E) }5&lt;/math&gt;<br /> <br /> [[2018 AMC 8 Problems/Problem 21|Solution]]<br /> ==Problem 22==<br /> Point &lt;math&gt;E&lt;/math&gt; is the midpoint of side &lt;math&gt;\overline{CD}&lt;/math&gt; in square &lt;math&gt;ABCD,&lt;/math&gt; and &lt;math&gt;\overline{BE}&lt;/math&gt; meets diagonal &lt;math&gt;\overline{AC}&lt;/math&gt; at &lt;math&gt;F.&lt;/math&gt; The area of quadrilateral &lt;math&gt;AFED&lt;/math&gt; is &lt;math&gt;45.&lt;/math&gt; What is the area of &lt;math&gt;ABCD?&lt;/math&gt;<br /> <br /> &lt;asy&gt;<br /> size(5cm);<br /> draw((0,0)--(6,0)--(6,6)--(0,6)--cycle);<br /> draw((0,6)--(6,0)); draw((3,0)--(6,6));<br /> label(&quot;$A$&quot;,(0,6),NW);<br /> label(&quot;$B$&quot;,(6,6),NE);<br /> label(&quot;$C$&quot;,(6,0),SE);<br /> label(&quot;$D$&quot;,(0,0),SW);<br /> label(&quot;$E$&quot;,(3,0),S);<br /> label(&quot;$F$&quot;,(4,2),E);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) } 100 \qquad \textbf{(B) } 108 \qquad \textbf{(C) } 120 \qquad \textbf{(D) } 135 \qquad \textbf{(E) } 144&lt;/math&gt;<br /> <br /> [[2018 AMC 8 Problems/Problem 22|Solution]]<br /> ==Problem 23==<br /> From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon?<br /> <br /> &lt;asy&gt;<br /> size(3cm);<br /> pair A[];<br /> for (int i=0; i&lt;9; ++i) {<br /> A[i] = rotate(22.5+45*i)*(1,0);<br /> }<br /> filldraw(A--A--A--A--A--A--A--A--cycle,gray,black);<br /> for (int i=0; i&lt;8; ++i) { dot(A[i]); }<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) } \frac{2}{7} \qquad \textbf{(B) } \frac{5}{42} \qquad \textbf{(C) } \frac{11}{14} \qquad \textbf{(D) } \frac{5}{7} \qquad \textbf{(E) } \frac{6}{7}&lt;/math&gt;<br /> <br /> [[2018 AMC 8 Problems/Problem 23|Solution]]<br /> <br /> ==Problem 24==<br /> In the cube &lt;math&gt;ABCDEFGH&lt;/math&gt; with opposite vertices &lt;math&gt;C&lt;/math&gt; and &lt;math&gt;E,&lt;/math&gt; &lt;math&gt;J&lt;/math&gt; and &lt;math&gt;I&lt;/math&gt; are the midpoints of edges &lt;math&gt;\overline{FB}&lt;/math&gt; and &lt;math&gt;\overline{HD},&lt;/math&gt; respectively. Let &lt;math&gt;R&lt;/math&gt; be the ratio of the area of the cross-section &lt;math&gt;EJCI&lt;/math&gt; to the area of one of the faces of the cube. What is &lt;math&gt;R^2?&lt;/math&gt;<br /> <br /> &lt;asy&gt;<br /> size(6cm);<br /> pair A,B,C,D,EE,F,G,H,I,J;<br /> C = (0,0);<br /> B = (-1,1);<br /> D = (2,0.5);<br /> A = B+D;<br /> G = (0,2);<br /> F = B+G;<br /> H = G+D;<br /> EE = G+B+D;<br /> I = (D+H)/2; J = (B+F)/2;<br /> filldraw(C--I--EE--J--cycle,lightgray,black);<br /> draw(C--D--H--EE--F--B--cycle); <br /> draw(G--F--G--C--G--H);<br /> draw(A--B,dashed); draw(A--EE,dashed); draw(A--D,dashed);<br /> dot(A); dot(B); dot(C); dot(D); dot(EE); dot(F); dot(G); dot(H); dot(I); dot(J);<br /> label(&quot;$A$&quot;,A,E);<br /> label(&quot;$B$&quot;,B,W);<br /> label(&quot;$C$&quot;,C,S);<br /> label(&quot;$D$&quot;,D,E);<br /> label(&quot;$E$&quot;,EE,N);<br /> label(&quot;$F$&quot;,F,W);<br /> label(&quot;$G$&quot;,G,N);<br /> label(&quot;$H$&quot;,H,E);<br /> label(&quot;$I$&quot;,I,E);<br /> label(&quot;$J$&quot;,J,W);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) } \frac{5}{4} \qquad \textbf{(B) } \frac{4}{3} \qquad \textbf{(C) } \frac{3}{2} \qquad \textbf{(D) } \frac{25}{16} \qquad \textbf{(E) } \frac{9}{4}&lt;/math&gt;<br /> <br /> [[2018 AMC 8 Problems/Problem 2|Solution]]<br /> ==Problem 25==<br /> How many perfect cubes lie between &lt;math&gt;2^8+1&lt;/math&gt; and &lt;math&gt;2^{18}+1&lt;/math&gt;, inclusive?<br /> <br /> &lt;math&gt;\textbf{(A) }4\qquad\textbf{(B) }9\qquad\textbf{(C) }10\qquad\textbf{(D) }57\qquad \textbf{(E) }58&lt;/math&gt;<br /> <br /> [[2018 AMC 8 Problems/Problem 25|Solution]]<br /> {{MAA Notice}}</div> Cosmicgenius